Do Tournaments Solve the Adverse Selection Problem?
Theofanis Tsoulouhas*
Ernest & Julio Gallo Management Program
School of Social Sciences, Humanities & Arts
University of California, Merced
Merced, CA 95343
October 16, 2013
Abstract. This paper provides a solution to a puzzle in the analysis of tournaments, that of why there
is no agent discrimination in practice. The paper examines the problem of a principal contracting with
multiple agents whose activities are subject to common shocks, when there is moral hazard and adverse
selection. The presence of common shocks invites the use of relative performance evaluation to minimize
the costs of moral hazard. But, in the additional presence of adverse selection, the analysis shows that at
the optimum there may be no need for ex ante screening through menus of contract o¤ers (i.e., for agent
discrimination). This is so because the principal becomes better informed ex post about agent types, via the
realization of common uncertainty, and can e¤ectively penalize or reward the agents ex post. Thus, unlike
the standard adverse selection problem without common uncertainty where the principal always bene…ts
from ex ante screening, it is shown that ex post sorting through relative performance evaluation reduces
the scope for ex ante screening through menus, and eliminates it completely if agents are known to not be
very heterogeneous. This is consistent with observed practice in industries where the primary compensation
mechanism is a cardinal tournament which is uniform among agents.
Keywords: Tournaments; Contests.
JEL Codes: D82, D86.
Acknowledgments: I am grateful to the participants of a workshop at Concordia University, Canada,
and the participants of the Society for the Advancement of Economic Theory conference at MINES ParisTech
in France for very useful comments.
1. Introduction
It is well established in contract theory that under adverse selection the uninformed party
can do better by o¤ering a menu of contract o¤ers to the informed party than by o¤ering a
single contract.1 Letting the agent self-select an o¤er from the menu enables the screening of
the agent’s type under appropriate incentive compatibility constraints. What is particularly
puzzling is the fact that in certain industries where the primary compensation mechanism
is relative performance evaluation via a cardinal tournament, uninformed parties o¤er a
uniform tournament contract to all the agents. What explains this empirical deviation, that
is, the lack of agent discrimination at the time of contracting, from the anticipated theoretical
optimum?
Considerable research on tournaments has been motivated by broiler contracting, ever
since Knoeber (1989) identi…ed the importance of this application. Cardinal tournaments
have been in use by processor companies during the production of broilers (in houses owned
and operated by growers) for decades now, and a lot of experience has been obtained with
the optimal design and …ne-tuning of such incentive schemes.2 One empirical regularity in
the production of broilers, which is quite puzzling, is the absence of agent discrimination at
the time of initial contracting or at the time of subsequent recontracting, even after some
information about grower abilities has been obtained. Ever since the work of Weitzman
(1980), Freixas, Guesnerie and Tirole (1985) and La¤ont and Tirole (1988), it has been
suggested that recontracting under more onerous terms once some information about ability
types is observed (i.e., the so called “ratchet e¤ect”) can distort agent incentives to perform
initially, which can partially explain the absence of agent discrimination at the time of recontracting.3 However, the absence of screening at the initial stage of contracting is extremely
puzzling. Lack of agent discrimination can also frequently be observed in the contracting of
HMOs with physicians, in sales contracts and in determining annual raises for faculty.
A cardinal tournament rewards an agent based on his performance relative to that of
his peers, speci…cally, agents receive a …xed base payment that induces participation, and a
variable bonus which depends on the di¤erence of an agent’s performance from the average
of his peer group. A cardinal tournament is a type of forcing contract that resolves the
moral hazard problem through penalties for performance below the agent group average and
rewards for performance above the group average. It also captures the essence behind a
1
For instance, see Page (1992, 1997). A classic example is insurance contracts, for instance automobile
insurance, where the insurance company proposes a menu to each applicant who selects an o¤er; the o¤ers
in the menu di¤er by the amount of the deductible.
2
For the stylized facts related to the production of broilers see Knoeber (1989), Knoeber and Thurman
(1994, 1995), Tsoulouhas and Vukina (1999) and Wu and Roe (2005, 2006).
3
See Allen and Lueck (1998) for a discussion of the rachet e¤ect in agriculture.
1
Crémer and McLean (1988) incentive scheme with correlated values. Starting with the seminal work of Lazear and Rosen (1981), Holmström (1982), Green and Stokey (1983), Nalebu¤
and Stiglitz (1983) and Malcomson (1984), ordinal tournaments (i.e., tournaments based on
rank) and cardinal tournaments have been shown to alleviate the moral hazard problem.4
In addition to moral hazard on the agent side, cardinal tournaments, speci…cally, have been
shown to alleviate the moral hazard problem on the principal side as well.5 However, do cardinal tournaments alleviate the adverse selection problem? Lazear and Rosen (1981) argue
that sorting between minor and major leagues would fail because agents would not self-select
(all agents would want to be in the major leagues, that is, in the high ability group). Given
that agents do not self-sort, agents in the resulting mixed leagues do not exert the e¢ cient
e¤ort. In this spirit, adverse selection due to agent heterogeneity has not received enough
attention in the tournament literature.6
This paper provides a solution to the puzzle of no agent discrimination at the time of
contracting which is traced to the very nature of tournaments (or relative performance evaluation) as opposed to the absolute performance evaluation contracts that standard contract
theory has considered. Speci…cally, tournaments screen and …lter away common production shocks from the responsibility of agents. Therefore, ex post relative performance of
an agent is a measure of an agent’s ability subject to his idiosyncratic (luck) component.
By contrast, ex post absolute performance of an agent is a measure of the agent’s ability,
realized common production shocks and idiosyncratic shock. In other words, to the extent
that common shocks are relatively signi…cant, ex post relative performance evaluation can
improve the accuracy of inferences made about agent ability. Thus, agents can be penalized
or rewarded ex post according to their inferred innate ability. The ex post ability to sort
agents, by obtaining informative signals about their abilities, and discriminate reduces the
scope for ex ante agent discrimination (screening) via contract menus. By contrast, without
tournaments, that is, with absolute performance evaluation, high agent performance would
be rewarded even if it were the sole outcome of pure stochastic e¤ects that are common
4
An associated literature is the one on contests, often based on contest success functions. See Tullock
(1980) for rent-seeking contests, Perez-Castrillo and Verdier (1992) for a more recent discussion of rentseeking games, Skaperdas (1996) for an axiomatization of contest sucess functions, Wärneryd (2003) for
uncertain prizes, and Konrad and Kovenock (2009) for contests with multiple rounds.
5
See Carmichael (1983) and Tsoulouhas (1999).
6
Early exceptions are O’Keefe, Viscusi and Zeckhauser (1984), Glazer and Hassin (1988), Bhattacharya
and Guasch (1988) and Yun (1997). O’Keefe, Viscusi and Zeckhauser show that the Lazear and Rosen
"climbing" problem can potentially be eliminated by increasing the prize spread, provided that the employer
monitors the contestants less precisely. Yun examines a modi…cation of the standard rank-order tournament,
that penalizes a small number of contestants heavily, to show that such a scheme can achieve …rst best
e¢ ciency when there is moral hazard and adverse selection with risk-neutral agents. More recently, Eriksson,
Teyssier and Villeval (2009) study the ex ante sorting e¤ect of tournaments and its impact on the variability
of e¤ort.
2
among the agents. Thus, the contribution of the paper is to show that whereas ex ante
agent discrimination through a screening contract makes perfect sense with absolute performance evaluation and in accordance with standard contract theory (for instance, see the
analysis in Bolton and Dewatripont (2005)), no ex ante discrimination via a single pooling
contract for each and every agent can be optimal under relative performance evaluation via
cardinal tournaments.7 The analysis characterizes an intuitive condition, which relies on
a measure of agent heterogeneity, under which a pooling contract o¤er to all the agents
dominates the o¤er of a screening contract menu. Speci…cally, the analysis shows that ex
post sorting through relative performance evaluation completely eliminates the scope for ex
ante screening through menus if agents are known to not be very heterogeneous. This is
consistent with the stylized facts for the broiler industry, at least, where growers (farmers)
share similar attributes. Thus, in all, tournaments do solve the adverse selection problem if
agents are not very heterogeneous. This result is in sharp contrast with standard contract
theory models without relative performance evaluation mechanisms where screening is always optimal regardless of agent heterogeneity. The degree of agent heterogeneity is very
important under tournaments in that ex ante screening is ine¢ cient if the agents are known
to not be very heterogeneous, given that tournaments enable the ex post sorting of agents.
The intuition is that whereas screening contracts can be more e¢ cient by tailoring
agent incentives to their innate abilities, such …ne-tuning comes at a monetary cost to the
principal over the pooling contract o¤er, especially given the principal’s ability to sort the
agents ex post via relative performance evaluation. In other words, why bother to gain more
information ex ante if you can get most of this information ex post at a lower cost? Only
if agents are su¢ ciently heterogenous does it make sense to engage in ex ante screening in
order to provide higher-power incentives for high ability types. If agents are not known to be
su¢ ciently heterogeneous, the pooling contract increases expected pro…ts because it does not
require the satisfaction of self-selection incentive compatibility constraints. Moreover, even
though the screening contract implements higher-power incentives for better ability types,
these types need to be paid a lot more in order to perform. In other words, higher-power
incentives for insigni…cantly better types is too costly.
The literature of adverse selection in tournaments is limited, however, our analysis is
related to several recent papers that examine agent heterogeneity. Riis (2010), examines
7
Regarding the second, associated puzzle, of why principals do not discriminate agents even after agent
types are learned through repeated observation of performance, in addition to the implications of a ratchet
e¤ect that were mentioned above, grower discrimination would add to their complaints about low pay and
subjection to group composition risk (see Tsoulouhas and Vukina (2001)). Besides, the strict requirements
of relationship-speci…c capital investments, in broiler production, and adherence to strict production rules
e¤ectively minimize the expression of higher ability into signi…cantly and systematically higher performance.
3
the e¢ ciency of contests among ex ante heterogeneous but risk-neutral agents, when the
prize structure can be modi…ed to adjust for the agents’strategic behavior, for instance, by
stipulating that the winner’s prize is determined by the identity of the runner up. Konrad
and Kovenock (2010) examine discriminating contests with stochastic abilities. Tsoulouhas
et al (2007) consider CEO contests that are open to heterogeneous outsiders.8
A word of caution is in order. The paper does not characterize overall optimal contracts or tournaments. Instead, it focuses on the speci…c puzzle of why there is no agent
discrimination under tournaments the way they are used in practice (at least in the case of
contracts for salesmen, agricultural contracts and contracts for physicians contracting with
HMOs). In these settings, standard cardinal tournaments are typically used. Prior research
has indicated that the overall optimal contract should be some partially non-linear tournament contract that depends on individual output relative to the average output obtained by
all agents (see, for instance, Tsoulouhas (1999), especially Appendix B therein).9 This is so
because in the presence of common shocks, individual performance is not a su¢ cient statistic
for individual e¤ort; by the strong law of large numbers, the output obtained by other agents
can provide an informative signal about the value of common shocks. Relative performance
evaluation constitutes a Pareto move because the principal uses the available information
more e¢ ciently. By removing common uncertainty from the responsibility of agents, and
by charging a premium for this insurance, the principal increases his pro…t without hurting
the agents. Moreover, by providing this type of insurance, tournaments enable the principal
to implement high power incentives. Tsoulouhas (1999) argues that the optimal contract
involves a likelihood ratio which is mostly linear (more so with a normal distribution than,
say, with a chi-square distribution). Signi…cant non-linearities only arise at very high output
levels, which are low probability events. However, Holmström and Milgrom (1987) have
argued that schemes that adjust compensation to account for rare events may not provide
correct incentives in ordinary high probability circumstances. By adhering to this premise,
Tsoulouhas (1999) and Tsoulouhas and Vukina (1999) characterized the standard cardinal
tournament, in which absolute performance and group average performance receive the same
weight, as a …rst-order Taylor series expansion of the optimal scheme at the average output
8
Tsoulouhas and Marinakis (2007), instead, analyze ex post agent heterogeneity.
Note some similarity with Al-Najjar (1997), even though Al-Najjar focused on the principal’s moral
hazard. He showed that this problem disappears as the number of agents grows, when conditioning on
the entire vector of outcomes improves the aggregation of information about the principal’s actions. AlNajjar focused on the optimality of statistical contracts motivated by Green (1990). Such contracts include
a payment scheme and a test statistic which depends on the vector of outcomes. If the value of the test
statistic is low, it generates massive information which allows the agents to appropriate the entire output.
In some sense, the test statistic in our analysis is the average output obtained by all agents who are subject
to common production shocks, when there is no moral hazard on the principal side, but there is adverse
selection and moral hazard on the agent side.
9
4
obtained by other agents. Subsequent to that, Tsoulouhas (2010) showed that "hybrid"
tournaments, that is, tournaments in which the weights on absolute and group average performances are not equal, are more e¢ cient. Such tournaments entail a partial …ltering of
common uncertainty. Providing for partial insurance against common uncertainty via a hybrid cardinal tournament is always better for the principal than providing for full insurance
against common uncertainty via a standard cardinal tournament. The principal can induce
the agents to exert more e¤ort by subjecting them to some common uncertainty. This is
because the agents work harder in order to insure themselves against bad realizations of the
common shocks. However, hybrid tournaments are more complicated to calculate and, to
the best of our knowledge, they are not used in practice. Further, in the limit, that is, for a
su¢ ciently large number of agents the weights on absolute individual and group average performance converge. Hence, in this paper the focus is on standard cardinal tournaments. The
additional advantage of using standard cardinal tournaments, in the context of the model
presented in Section 2, is that one is able to not only obtain clean closed form solutions for
the contractual variables but, in addition, the solutions highlight the importance of parameters such as the variances of idiosyncratic versus common production shocks, as well the
variance of agent ability.
But what about ordinal tournaments? Even though the founders of tournament theory,
Lazear and Rosen (1981), focused on ordinal or rank-order tournaments, such tournaments
are informationally wasteful when data on the agents’ cardinal performance are available
(following Holmström (1982)), which is certainly the case with broiler production. Moreover,
Tsoulouhas (2012) has shown that, contrary to anecdotal popular belief, switching from
ordinal to cardinal tournaments improves e¢ ciency. Lastly, with rank-order tournaments
and screening menus, each agent would have to self-select into di¤erent sets of prespeci…ed
prizes, which is arguably very di¢ cult to determine and, hence, to the best of our knowledge,
is non-existent in practice.
Again, Section 2 presents the model. Section 3 examines the perfect information tournament as a benchmark for the analysis. Section 4 presents the case when the ability type
of each agent is privately known to the agent, but the principal pools the agents by o¤ering
the same tournament contract to each of them without screening. Section 5 examines the
case in which the principal screens the agents by o¤ering them a menu and letting them
self-select. Section 6 contrasts the pooling to the screening contract to obtain the core result
in the paper. Section 7 concludes.
2. The model
A principal signs a contract with a …nite number of agents, n. The number of agents
5
is assumed to be determined exogenously.10 Each agent i produces observable output xi
according to the production function xi = ai + ei + + "i , where ai is the agent’s …nite
ability, ei is the agent’s …nite e¤ort, is a common shock in‡icted on all agents and "i is an
idiosyncratic shock. Agents know their own ability types at the time of contracting, but the
principal does not. In addition to adverse selection there is moral hazard, in that each agent’s
e¤ort and the subsequent realizations of the production shocks are private information to
him, but the output obtained is publicly observed.11 The price of output is normalized to 1.
Each agent’s type, ai ; independently follows a normal distribution with mean and …nite
variance 2a . Independently of ai , the common shock follows a normal distribution with
mean zero and variance 2 and, independently of ai and , the idiosyncratic shock "i follows
a normal distribution with zero mean and …nite variance 2" . Because cardinal data on the
performance of each agent are assumed to be available, the principal compensates agents
for their e¤orts based on their outputs by using a cardinal tournament.12 The cardinal (or
two-part piece rate) tournament is the payment scheme in which the compensation wi to
each agent i is determined by relative performance. Speci…cally,
wi = b + (xi
x) = b +
n
1
n
xi
1P
xj
n j6=i
!
;
(1)
where x is the average output obtained by all agents, b is the base payment and is the
bonus factor to be determined by the principal.13 Note that under the tournament the total
wage bill is proportional to the base payment b, that is, wi = nb: Thus, the principal’s
total payment to the agents and, hence, the expected payment per agent are independent
of output (Ewi = b), however, each agent’s relative performance determines his share of the
…xed total payments.
Agent preferences are represented by a CARA utility function of the form u(wi ; ei ) =
exp rwi + 2r e2i ; where r 6= 0 is the agent’s coe¢ cient of absolute risk aversion. Given
the normality assumptions above, both xi and wi are normally distributed and, therefore,
u( ) follows a lognormal distribution. An important property of the lognormal distribution
is that its …rst moment has a closed form, which allows us to obtain analytical solutions for
10
See Myerson and Wärneryd (2006) for a model where the set of players is a random variable.
Note that tournaments alleviate the principal’s moral hazard problem in providing lower quality inputs
to the agent (see Carmichael (1983) and Tsoulouhas (1999)).
12
Rank-order (ordinal) tournaments that ignore the agents’cardinal performance are informationaly wasteful in this case (Holmström (1982)).
13
As argued earlier, Tsoulouhas (2010) has shown that "hybrid" cardinal tournaments of the form wi =
b + xi
x are dominant over standard tournaments of the form wi = b + (xi x), however, the former
are more complicated to calculate and, to the best of our knowledge, they are not used in practice. Further,
in the limit, that is, for a su¢ cinetly large number of workers, = :
11
6
the contractual parameters.14 Another advantage of this model is that the baseline results
that can be obtained with homogeneous agents conform with those obtained by Lazear and
Rosen (1981) who, even though they utilized more general utility functions, relied on …rstorder Taylor approximations of these functions. For ease of exposition, we normalize each
agent’s reservation utility to 1.15 Thus, we assume that whereas each agent knows his true
ability level, the "market" does not, therefore, all agents have the same opportunity cost
and, hence, the same "reservation" utility.16
3. The perfect information tournament
This section analyzes a benchmark for the analysis, which is the tournament if the type of
each agent were known to the principal and all agents. If the agent’s type ai were known,
the contractual parameters o¤ered by the principal to each agent would depend on his type,
and the principal would determine these parameters (b(ai ); (ai )) by backward induction.
Under a tournament w(ai ) = b(ai ) + (ai )(xi x); the agent’s expected utility after he signs
the contract but before exerting e¤ort is
(2)
EU (ei (ai )jai ; b(ai ); (ai )) =
=
exp
(
r
"
b(ai ) + (ai ) nn 1 (ai + ei (ai )
(ei (ai ))2
2
aj )
(ai ) n1
r( (ai ))2 n 1
2
n
2
"
P
j6=i ej (aj )
#)
;
where the expression in the square brackets is the certainty equivalent compensation for the
P
agent of type ai who has signed contract (b(ai ); (ai )), and aj =
1) is the
j6=i aj =(n
average type of all agents other than i who are also participating in the tournament.17 Note
that when agent types are known, expectations are only over production uncertainties. Also
observe that expected utility rises with increases in the expected payment from the principal,
reductions in the e¤ort level implemented by the principal and reductions in the variance
of the payments. By backward induction, to ensure the compatibility of the contract with
agent incentives to perform, the principal calculates the e¤ort level that maximizes (2). First
order conditions yield
n 1
(ai );
(3)
ei (ai ) =
n
2
This is so because E[exp( rwi + 2r e2i )] = exp[m + 2 ]; when rwi + 2r e2i s N (m; 2 ); which allows us
to obtain a closed form solution for the expected utility.
15
Note that the analysis is directly applicable to any normalization other than 1.
16
Note that a similar assumption is made in Bolton and Dewatripont (2005, p. 53), in that the reservation
utilities are normalized to zero for all agents. When the reservation utilities are type independent, there is
no need to worry about countervailing incentives, which are present when the principal has to o¤er such a
great deal to highly able agents in order to attract them that makes low ability agents mimic the high types.
17
Given the modeling assumptions the certainty equivalent admits a mean-variance form.
14
7
which implies that incentives to perform are fully determined by the incentives provided
by the principal via the bonus factor (ai ). Note that ability a¤ects the e¤ort choice of
the agent through the choice of the optimal bonus factor (ai ).18 Given (3), to ensure the
compatibility of the contract with agent incentives to participate, the principal selects the
value of the base payment, b(ai ), that satis…es the agent’s individual rationality constraint
with equality so that the agent receives no rents but still accepts the contract. Hence, the
individual rationality constraint of a type ai agent is:
EU (b(ai ); (ai )jai ; ei (ai )) =
=
exp
r b(ai ) + (ai )
() b( (ai )jai ) =
n
1
n
(ai
1n 1
2 n
1n 1
2 n
aj )
n
1
n
+r
2
"
n
1
n
+r
n
( (ai ))2
( (ai ))2
2
"
1
n
(ai
=
1 ()
aj ) (ai ):
(4)
By setting b in accordance with (4) the principal can ensure agent participation at least cost.
That is, the individual rationality constraint is binding and yields no rents to all agent types.
Given conditions (3) and (4), the principal maximizes expected total pro…t
E
PI
( (ai )) =
n
P
[Exi
Ewi ] = n [a + ei (ai )
b( (ai )jai )] =
(5)
i=1
=n a+
n
1
n
(ai )(1 + ai
1n 1
2 n
aj )
n
1
n
+r
2
"
( (ai ))2 ;
P
where a = i ai =n is the average type of all competing agents. The solution to the principal’s
maximization problem satis…es
(ai ) =
1 + (ai aj )
; 8ai ;
n 1
+ r 2"
n
(6)
that is, the bonus factor depends positively on the agent’s ability ai relative to the average
ability aj of other agents, because the principal implements higher-power incentives for
relatively stronger agents. However, the larger the variance of the idiosyncratic shock, 2" ,
the lower the bonus factor, (ai ), because the weaker the link between the power of incentives
and output. In order to provide correct incentives to the agents, note that the principal will
hire an agent of type ai only if ai > aj 1: This condition is equivalent to ai > a nn 1 ; 8i; in
other words, the principal picks a group of agents from the population so that the ability ai
18
Ability would also a¤ect the e¤ort choice directly if the cost of e¤ort was dependent on ability, but in
this case the analysis would be intractable. This is because the agent’s compensation inclusive of monetized
e¤ort costs would not follow a normal distribution, hence, utility would not follow a lognormal distribution.
8
of each agent in the group is larger than the average ability a of all the agents in the group
minus nn 1 : The condition is clearly satis…ed if it holds for the lowest ability agent in the
group. Given condition (6), condition (4) implies
1n 11
b(ai ) =
2 n
(ai aj )2
1 1 (ai aj )2
=
; 8ai :
n 1
2 1 + nn 1 r 2"
+ r 2"
n
(7)
In contrast to the bonus factor (ai ), the base payment b(ai ) depends negatively on ability
ai relative to the average ability aj of other agents, because relatively more able agents need
weaker incentives to participate since the likelihood of them receiving a bonus payment is
higher. Moreover, the less ability ai di¤ers from average ability aj of other agents regardless
of direction, that is, the more uniform the group is, the larger the base payment b(ai ) needed
to induce agent participation because the less likely it is for each agent to receive a bonus.
Moreover, assumption ai > aj 1 implies that b(ai ) < 0; 8i (if ai is not smaller than
aj + 1):19 Note that neither the bonus factor nor the base payment depend on 2 because
the principal …lters away common shocks from the responsibility of the agents. Obviously,
neither payment should depend on the variance of ability either, because abilities are assumed
to be known in this benchmark. Given conditions (6) and (7), the principal’s expected pro…t
under tournament is
E
PI
=
n
P
i=1
a+
n
1
n
(ai )
b(ai ) =
n
P
i=1
1 n 1 (1 + ai aj )2
a+
:
n 1
2 n
+ r 2"
n
(8)
Condition (8) indicates that the principal’s expected pro…t increases with the average ability
of all agents and the relative ability of each agent, the latter having a positive impact on
incentives and a negative impact on the expected payment per agent as measured by the
base payment.
4. The tournament without screening
Next we turn to the case when the ability type ai of each agent is privately known to the
agent, but the principal pools the agents by o¤ering the same tournament contract (b; ) to
each of them without screening, so that wi = b + (xi x). Whereas each agent knows his
type, he does not know the type of any other agent which, given independent ability draws,
he expects to be equal to E(aj jai ) = E(aj ). The agent’s expected utility after he signs the
19
Note that the agents will still participate because (7) is obtained by (4). This is feasible because the
agents are not liquidity constrained. The optimality of cardinal tournaments under liquidity constraints has
recently been examined by Marinakis and Tsoulouhas (2012). The optimality of tournaments under limited
liability for the principal has been examined by Marinakis and Tsoulouhas (2013).
9
contract, but before exerting e¤ort, is
=
exp
(
EU (ei jai ; b; ) =
"
n
r b+
1
n
(ai + ei
(9)
1P
ej
n j6=i
E(aj ))
e2i
2
2
r
n
2
2
a
1
n
n
+
2
"
#)
:
The principal calculates the e¤ort level that maximizes (9). First order conditions yield
ei =
n
1
n
(10)
:
The individual rationality constraint of a type ai agent is:
EU (b; jai ; ei ) =
=
exp
r b+
n
1
n
() b( jai ) =
(ai
1n 1
2 n
E(aj ))
n
1
n
1n 1
2 n
2
a
+r
n
+
n
1
n
+r
n
n
2
2
"
2
a
+
1
n
2
2
"
(ai
=
1 ()
E(aj )) :
(11)
But which ability type should be considered? Given the assumption that the number of
agents n is exogenously determined, the principal will set ai = e
a so that the required n
agents of ability ai e
a are expected to accept the contract o¤er, that is, Pr(ai e
a)N = n;
where N is the total number of available agents. Note that by setting ai = e
a, the individual
rationality constraint is non-binding for all types ai > e
a, because expected utility is increasing
in ability ai (see (9) above). Thus, this result is di¤erent from the First Best case where all
agent types would get no rents, because the principal in the current case makes a single o¤er
which must be expected to be accepted by n agents.
Given conditions (10) and (11)), the principal maximizes expected total pro…t
E
=
Pr(aP
a)N
i e
[Exi
Ewi ] = Pr(ai
i=1
= n E(ai ) +
n
1
n
(1 + e
a
P
e
a)N [E(ai ) + ei
E(aj ))
(12)
( )=
1n 1
2 n
b( je
a)] = n [E(ai ) + ei
n
1
n
+r
2
a
n
+
2
"
b( je
a)] =
2
;
where E(ai ) is the expected agent ability that is formed by the principal for agents who
accept the contract o¤er. However, the principal and rational agents who receive independent
ability draws should form the same expectation so that E(ai ) = E(aj ): The solution to the
10
principal’s maximization problem, then, satis…es
=
and
b=
1 n 1 1 (e
a
2 n n 1 +r
n
1 + (e
a
n 1
n
+r
E(ai ))2
2
a
n
+
E(ai ))
2
a
n
=
2
"
(13)
;
2
"
+
1 1 (e
a E(ai ))2
:
2 1 + n r 2a + 2
"
n 1
n
(14)
The principal will o¤er (b; ) to all agents, as determined by (14) and (13). Note that
the bonus factor depends positively on the "cut-o¤" ability e
a relative to the expected
ability E(ai ), because the principal implements higher-power incentives for relatively stronger
agents, but the base payment b depends negatively on the "cut-o¤" ability e
a relative to
the expected ability E(ai ), because relatively more able agents need weaker incentives to
participate. Similar to the benchmark perfect information case, we assume that the number
of agents and the distribution of ability are such that e
a > E(ai ) 1, in order for the agents to
be provided with correct incentives. Unlike the benchmark case though, parameters b and
depend negatively on the variance of ability, 2a , because the higher the variance of ability the
weaker the link between the power of incentives and output. Moreover, the less the "cut-o¤"
ability e
a di¤ers from expected ability E(ai ), regardless of direction, that is, the more uniform
the group is, the larger the base payment b needed to induce agent participation, because
the less likely it is for each agent to receive a bonus. Given (13) and (14), the principal’s
expected pro…t under tournament is
E
P
= n E(ai ) +
n
1
n
2
b = n 4E(ai ) +
1 n 1 (1 + e
a
2 n n 1 +r
n
2
3
E(ai )) 5
:
2
a
2
+
"
n
(15)
Condition (15) indicates that the higher the expected agent ability E(ai ) and the higher the
"cut-o¤" ability e
a relative to the expected agent ability the higher the principal’s expected
pro…t. Note that, as Appendix A shows,
E(ai ) =
e
aF (e
a) + lim
! 1
Z
e
a
F (ai )dai :
(16)
5. The tournament with screening
We now turn to the case when the ability type of each agent is privately known to the agent
but the principal screens the agents by o¤ering them a menu fb(ai ); (ai )g instead of a single
o¤er. The menu provides each agent with incentives to self-select the o¤er in the menu that
11
is designed for his true type, so that the wage for agent type ai is w(ai ) = b(ai )+ (ai )(xi x).
Thus, the principal is completely uninformed and expects any agent’s ability to be equal to
before the agent signs the contract, but knows the true ability ai of each agent after the
agent signs the contract. Whereas each agent knows his own type, he does not know the
type of any other agent which, given independent ability draws, he expects to be equal to
before and after signing the contract. The principal still determines the optimal contractual
parameters by backward induction. The agent’s expected utility after he self-selects and
signs an o¤er (b(ai ); (ai )) in the menu, but before exerting e¤ort, is
(17)
EU (ei (ai )jai ; b(ai ); (ai )) =
=
exp
(
r
"
b(ai ) + (ai ) nn 1 (ai + ei (ai )
(ei (ai ))2
2
(ai ) n1
)
r( (ai ))2 n 1
2
n
2
a
n
+
2
"
P
j6=i ej
#)
:
By backward induction, to ensure the compatibility of the contract with agent incentives to
perform, the principal calculates the e¤ort level that maximizes (17). First order conditions
yield
n 1
ei (ai ) =
(ai ); 8ai :
(18)
n
Given (18), to ensure the compatibility of the contract with agent incentives to participate, the principal selects the value of the base payment, b(ai ), that satis…es the individual
rationality constraint of type ai agent:
EU (b(ai ); (ai )jai ; ei (ai )) =
=
exp
() b( (ai )jai )
(
r
"
1n 1
2 n
b(ai ) + (ai ) nn 1 (ai
n 1
n
1n 1
2 n
n
1
n
+r
2
a
+r
n
2
a
n
+
+
2
"
)
2
"
(ai )2
( (ai ))2
#)
1 ()
n
1
n
(ai
) (ai ):
(19)
Thus, a base payment b( (ai )jai ) that satis…es condition (19) is individually rational.
By appealing to the Extended Revelation Principle, in order to ensure the compatibility
of the contract with each agent type’s incentives to select the menu o¤er that is designed
for his true type, the principal needs to make sure that the following additional incentive
compatibility constraint is satis…ed for every type ai :
ai = arg max EU (b(abi ); (abi )jai ; ei (abi )) =
abi
12
(20)
=
exp
r b(abi ) + (abi )
n
1
(ai
n
1n 1
2 n
)
n
1
n
2
a
+r
n
+
2
"
The following Lemma is applicable to this constraint.
( (abi ))2
:
Lemma 1 The incentive compatibility constraint (20) is equivalent to:
db(ai )
+ (ai
dai
n
)
1
2
a
+r
n
2
"
+
n
d (ai )
dai
n
(ai )
1 d (ai )
= 0; 8ai ;
n
dai
(21)
0; 8ai ;
(22)
where constraint (21) is the local incentive compatibility constraint and constraint (22) is the
monotonicity constraint.
Proof. The F.O.C. of the maximization problem in (20) imply
db(abi )
+ (ai
dai
n
)
1
n
2
a
+r
+
n
2
"
(abi )
is satis…ed 8ai = abi :
The S.O.C. of the maximization problem imply
d2 b(abi )
+ (ai
da2i
n
)
n
1
n
1
+r
n
2
a
+r
n
2
"
+
2
a
n
2
"
+
n
(abi )
d (abi )
dai
1
n
n
2
+
n
1 d (abi )
=0
n
dai
1 d2 (abi )
n
da2i
n
1 d (abi )
n
dai
(23)
(24)
0
is satis…ed 8ai = abi :
First note that the F.O.C. above implies condition (21). Second, observe that di¤erentiating (21) with respect to ai implies
d2 b(ai )
+ (ai
da2i
)
n
1
n
+r
n
1
n
2
a
n
+
+r
2
a
n
+
n
2
"
1
n
2
"
(ai )
d (ai )
dai
n
1 d2 (ai )
n
da2i
2
+2
n
(25)
1 d (ai )
= 0
n
dai
The latter condition and the S.O.C. imply
n
1 d (ai )
n
dai
0;
(26)
which in turn implies (22). Thus, so far we have proved that the incentive compatibility
13
constraint (20) implies the local incentive compatibility constraint (21) and the monotonicity
constraint (22).
Next, suppose that both the local incentive compatibility and the monotonicity constraints (21) and (22) hold. We need to show that the incentive compatibility constraint
(20) is satis…ed. Suppose that there exist ai such that the local incentive compatibility
constraint (21) and the monotonicity constraint (22) hold, but the incentive compatibility
constraint (20) is violated for at least one abi 6= ai . Thus,
exp
r b(ai ) + (ai )
n
1
n
(ai
1n 1
2 n
)
n
1
n
2
a
+r
n
+
( (ai ))2
2
"
<
(27)
<
r b(abi ) + (abi )
exp
n
1
n
(ai
1n 1
2 n
)
n
1
n
+r
2
a
n
+
2
"
By integrating we obtain
Zabi
exp
r
db( i )
+ (ai
d i
n
)
1
n
2
a
+r
n
2
"
+
( i)
n
( (abi ))2
1 d ( i)
n
d i
d
i
:
> 0:
ai
(28)
Constraint (22) implies
<
d ( i)
d i
(ai
)
(
)
i
0: Assuming abi > ai ; it follows that
n
1
n
n
1
n
2
a
+r
n
2
a
+r
n
+
2
"
+
2
"
1 d ( i)
<
n
d i
n 1 d ( i)
( i)
:
n
d i
( i)
n
(29)
Hence, (21) implies
Zabi
exp
r
db( i )
+ (ai
d i
n
)
1
n
+r
2
a
n
2
"
+
( i)
n
1 d ( i)
n
d i
d
i
< 0;
ai
(30)
which contradicts (28). A similar result can be obtained if abi < ai .
Therefore, in order to characterize the optimal menu fb(ai ); (ai )g; with w(ai ) = b(ai ) +
(ai )(xi x); and given condition (18), the principal maximizes expected total pro…t
E
S
(b(ai ); (ai )) =
n
P
[Exi
Ewi ] =
i=1
=
n
P
i=1
n
P
[ + ei (ai )
i=1
+
n
1
n
14
(ai )
b(ai ) ;
b(ai )] =
(31)
subject to the individual rationality constraint (19), the local incentive compatibility constraint (21) and the monotonicity constraint (22).
But, in the presence of the incentive compatibility constraint (21), is the individual rationality constraint (19) binding or non-binding? The individual rationality constraint would
be binding for every ai if b( (ai )jai ) could be reduced for every ai so that (19) held as an
equality without hurting the local incentive compatibility constraint (21) or the monotonicity constraint (22). But reducing b( (ai )jai ) so that (19) holds as an equality for every
ai may require a non-parallel shift in the b(ai ) schedule which could, in principle, a¤ect
db(ai )
: The following Lemma shows that b( (ai )jai ) can be reduced so that (19) holds as
dai
an equality without hurting the local incentive compatibility constraint (21), assuming that
the monotonicity constraint is non-binding. Thus, the individual rationality constraint is
binding.
Lemma 2 Assuming that the monotonicity constraint (22) is non-binding for every ai ; the
individual rationality constraint (19) is binding for every ai :
Proof. First note that without the individual rationality constraint (19), the local incentive
compatibility constraint (21) is binding. This is so because the objective function without the
individual rationality constraint (19) or the local incentive compatibility constraint (21) is
strictly increasing (and linear) in (ai ) and strictly decreasing (and linear) in b(ai ): Therefore,
the optimal (ai ) is the highest possible, for every ai ; with (ai ) > 0 in order to provide
correct incentives to the agent and (ai ) < +1 because of the assumption of …nite e¤ort, and
the optimal b(ai ) ! 1; for every ai : This solution violates the local incentive compatibility
constraint (21).
Second, without the individual rationality constraint (19), by substituting the binding
local incentive compatibility constraint (21) into the objective function, the maximization
problem reduces to choosing b(ai ) and (ai ) to maximize
n
P
i=1
2
4 +n
1
n
h
b0 (ai )
n 1 0 (ai )
n
n 1
n
+r
+ (ai
2
a
n
+
)
2
"
i
3
b(ai )5 ;
(32)
which is strictly decreasing (and linear) in b(ai ) and is independent of (ai ): Thus, b(ai ) !
1; 8ai ; which violates the individual rationality constraint (19) for every ai ; given a …nite
number of agents.
The result in Lemma 2 is in sharp contrast with standard adverse selection theory models
without tournaments (for instance, see Bolton and Dewatripont (2005), pp. 47-56, 77-81
15
and 82-88) where the individual rationality constraint of all types other than the lowest is
non-binding. Here, agents need not be given information rents in order to participate and
select the o¤er in the menu fb(ai ); (ai )g that was designed for their true ability type. The
rationale is that the principal becomes better informed ex post about agent types through the
realization of common uncertainty which he …lters through the average output x obtained
by the agents. By the strong law of large numbers x does provide an informative signal
about the value of common shocks. Thus, the principal can e¤ectively penalize or reward
the agents ex post without the need for providing them with excessive incentives ex ante.
Further, as the next lemma indicates, the local incentive compatibility constraint is also
binding for almost all types but one.
Lemma 3 Assuming that the monotonicity constraint (22) is non-binding for every ai ; the
local incentive compatibility constraint (21) is binding for every ai 6=
1, and is non-binding
for ai =
1:
Proof. First note that, without the local incentive compatibility constraint (21), the individual rationality constraint is binding because the principal will …nd it optimal to reduce
b(ai ); 8ai ; so that the agent receives no rents. The solution then satis…es conditions similar
to (7) and (6) under perfect information, but with instead of aj ; because the average type
of agents is not known to the principal and the agents in this case, which also introduces
uncertainty about the types of other agents captured by 2a . Hence,
(ai ) =
and
b(ai ) =
1 + (ai
n 1
n
+r
1
1
21+
n
)
2
a
n
)2
(ai
n
r
1
; 8ai ;
2
"
+
2
a
n
; 8ai :
2
"
+
(33)
(34)
By di¤erentiating (19) with equality with respect to ai we obtain:
db(ai )
=
dai
=
n
1
n
n
=
1
n
n
1
n
+r
+r
2
a
n
2
a
n
2
"
+
+
2
"
(ai )
(ai )
(35)
d (ai )
dai
(ai
16
n
1
n
)
n
(ai
)
1 d (ai )
n
dai
d (ai )
dai
n
n
n
1
n
1
(ai ) =
(ai ); 8ai :
Therefore,
db(ai )
+ (ai
dai
=
n
1
n
2
a
+r
n
)
n
+ (ai
(ai )
1
n
=
n
1
(ai )
n
)
2
a
+r
n
2
"
+
(ai
n
)
2
a
+r
n
2
"
+
1
1 d (ai )
n
dai
2
"
+
n
n
1 d (ai )
=
n
dai
n
1
n
(ai )
n
(36)
(ai )+
1 d (ai )
=
n
dai
(ai ); 8ai :
n
Given (33), it follows that
db(ai )
+ (ai
dai
n
)
1
n
n
=
2
a
+r
1
2
"
+
n
1 + (ai
n
n 1
n
)
2
a
+r
(ai )
n
2
"
+
n
1 d (ai )
=
n
dai
(37)
; 8ai :
However,
n
1
n
1 + (ai
n 1
n
)
2
a
+r
n
+
(38)
=0
2
"
only if ai =
1: Thus, the the local incentive compatibility constraint (21) is automatically
satis…ed and, hence, non-binding if ai =
1; and is binding otherwise.
Lemma 3 indicates that without the local incentive compatibility constraint (21), if agents
were o¤ered a menu fb(ai ); (ai )g with b(ai ) satisfying (34) and (ai ) satisfying (33), only
type ai =
1 would trivially self-select the o¤er meant for his type.
Given condition (18), and given Lemmas 2 and 3, for ai 6=
1; the principal maximizes
expected total pro…t subject to a binding individual rationality constraint (19) with equality,
and a binding local incentive compatibility constraint (21), assuming that the monotonicity
constraint (22) is non-binding. Note that (21) implies
db(ai )
+ (ai
dai
)
n
1
n
()
+r
(ai ) =
2
a
n
h
+
2
"
b0 (ai )
n 1 0 (ai )
n
n 1
n
+r
17
(ai )
n
+ (ai
2
a
n
+
2
"
1 d (ai )
= 0; 8ai ;
n
dai
i
)
; 8ai ;
()
(39)
db(ai )
where b0 (ai )
and
dai
and (ai ) to maximize
E
S
0
d (ai )
:
dai
(ai )
(b(ai ); (ai )) =
n
P
Hence, for ai 6=
[Exi
1; the principal chooses b(ai )
Ewi ] = n [ + ei (ai )
(40)
b(ai )] =
i=1
=
n
P
+
n
=
n 6
P
6
4
i=1
n 1
n
+
1n 1
2 n
n 1
n
2
a
+r
n
(ai )
n
i=1
2
1
0
n b (ai )
+(ai
n 1 0 (ai )
2
n 1
+r na + 2"
n
2
"
+
b(ai ) =
3
)
n 1
n
( (ai ))2 +
(ai
) (ai )
which leads to the following Lemma.
7
7;
5
Lemma 4 Assuming that the monotonicity constraint (22) is non-binding for every ai , the
contract menu fb(ai ); (ai )g; with wi (ai ) = b(ai ) + (ai )(xi x); satis…es
(ai ) =
b(ai ) =
(ai
n 1
n
)
2
a
+r
1n 1
2 n n
n
)2
(ai
1
n
+r
; 8ai > ;
2
"
+
2
a
n
2
"
+
(41)
; 8ai > ;
(42)
and
(ai ) = 0; 8ai
;
(43)
b(ai ) = 0; 8ai
:
(44)
Proof. The F.O.C. of the maximization problem above satisfy
n
1
n
n
1
n
+r
2
a
n
+
hence
(ai ) =
2
"
(ai ) +
(ai
n 1
n
+r
18
)
2
a
n
+
n
1
n
:
2
"
(ai
) = 0;
(45)
Therefore, (19) implies,
1n 1
b(ai ) =
2 n
n
1
n
n
1
n
(ai
+r
)
1n 1
2 n n
=
2
a
n
(ai
n 1
n
n
+r
n 1
n
)
2
a
+r
2
a
n
2
"
+
2
=
+
n
+r
2
"
2
(ai
1
+
)2
(ai
2
"
)
2
a
n
:
2
"
+
Conditions (33) and (34) prove (43) and (44) for ai =
1. Nevertheless, if ai
the
pro…t maximizing principal would always o¤er (ai ) = 0 and b(ai ) = 0 to provide correct
incentives.
In the remaining analysis, and in accord with section 4 above, we assume that Pr(ai > )N
n; that is there are enough agents whose type is above the population average. Lemma 5
below completes the analysis by noting that the monotonicity constraint, which was assumed
to be non-binding earlier, is indeed non-binding.
Lemma 5 The monotonicity constraint (22) is non-binding for every ai > :
Proof. Given condition (41), it follows that
d (ai )
=
dai
1
n 1
n
2
a
+r
n
> 0;
+
2
"
which completes the proof.
Given the analysis above, the principal’s expected pro…t under tournament with screening before agents accept the contract is
E
S
=
n
P
i=1
+
n
1
n
(ai )
b(ai ) =
n
P
i=1
2
4 +n
1 (ai
n
n 1
n
1
2
) + (ai
+r
2
a
n
+
)
2
"
2
3
5:
(46)
6. The dominant tournament
In this section we determine the dominant tournament when the ability type of each agent is
privately known to the agent, given equations (15), with e
a > E(ai ) 1 and Pr(ai e
a)N = n;
where N is the total number of available agents, and (46), with ai > and Pr(ai > )N n.
The proposition that follows compares the screening to the pooling tournament.
19
Proposition 6 Assuming e
a > E(ai ) 1 and ai > ; the screening menu of contract o¤ers
fb(ai ); (ai )g; with w(ai ) = b(ai ) + (ai )(xi x); dominates the pooling contract o¤er (b; );
with wi = b + (xi x); from the principal’s perspective i¤
Pn
e
a + A) [1 + (e
a E(ai ) + 1) + (ai
n
i=1 (ai
where E(ai ) satis…es condition (16) and A
)]
1 + 2A
E(ai )
(n
1) + r ( 2a + n 2" )
;
(n 1)
(47)
:
Proof. Equations (15) and (46) yield:
E
()
()
n
P
(ai
i=1
n
P
1 + 2(e
a
1
) + (ai
2
ai
)
A) + (e
a
2
S
nA
P
E
n 1
n
()
2
a
+r
n
(48)
+
n 1
n
2
A)
(ai
2
)
2
"
1
+ n (1 + e
a
2
2nA
n 1
n
()
(ai
i=1
e
a + A) [1 + (e
a
E(ai ) + 1) + (ai
)]
n + 2nA
2
a
+r
n
+
2
"
()
n 1
n
i=1
n
P
A)2 ()
n 1
n
+r
2
a
n
n 1
n
+
2
"
;
which completes the proof.
Equation (47) provides the main result of the paper. Observing that the LHS of the equation
is a crude measure of variability in agent types that accept the contract o¤ers, the equation is
stating that when agent type variability is relatively small, the principal does not bene…t by
o¤ering a menu of contracts to screen the agents. That is, expected pro…t for the principal
is greater under a uniform or pooling contract o¤er to all agents, provided that agent ability
is known to not vary much. Only if agent ability varies signi…cantly (i.e., it is larger than a
threshold) it is pro…table for the principal to screen the agents in order to provide higherpower incentives to more able agents.
7. Conclusions
The result obtained in section 6, stating that when agent type variability is small the principal does not bene…t by o¤ering a menu of contracts to screen the agents, is in sharp
contrast with standard adverse selection theory models without tournaments (for instance,
see Bolton and Dewatripont (2005), pp. 47-56, 77-81 and 82-88). The rationale is that,
under tournaments, and regardless of whether the principal makes a pooling or a screening
o¤er, the principal becomes better informed ex post about an agent’s ability type through
20
the realization of common uncertainty which he …lters through the average output x obtained by the agents. By the strong law of large numbers x provides an informative signal
about the value of common shocks. Thus, the principal can e¤ectively penalize (reward)
the agents ex post through a reduced (increased) ex post monetary reward. The feasibility
of penalizing an agent ex post through a tournament reduces the scope for screening agent
types ex ante through the o¤er of a menu they have to chose from, even though it facilitates
the implementation of screening contracts without the need for paying excessive information
rents. Provided that agent ability is known to not vary much among agents, if the principal
observes a better performance, he interprets it as coming from idiosyncratic luck and there
is no need to reward luck in order to provide correct incentives. Only if agents are known
to be su¢ ciently heterogeneous, then it is optimal for the principal to screen the agents
with individualized o¤ers through a menu. In that case, when the principal observes a high
performance, it is more likely that it is coming from a more able type who must be rewarded
accordingly through an individualized o¤er. Further, if agents are not known to be su¢ ciently heterogeneous, the pooling contract increases expected pro…ts because it does not
require the satisfaction of self-selection incentive compatibility constraints. Moreover, even
though the screening contract implements higher-power incentives for better ability types,
these types need to be paid a lot more in order to perform, which reduces the principal’s
pro…t. In all, the intuition is that whereas screening contracts can be more e¢ cient by tailoring agent incentives to their innate abilities, such …ne-tuning comes at a monetary cost to
the principal over the pooling contract o¤er, especially given the principal’s ability to sort
the agents ex post via relative performance evaluation. In other words, why bother to gain
more information ex ante if you can get most of this information ex post at a lower cost?
The analysis of this paper provides a solution to a puzzle in tournament theory, that of
why there is no ex ante agent discrimination in practice. Unlike standard contract theory
where, regardless of agent heterogeneity, the uninformed party can do better through agent
discrimination rather than by o¤ering a single contract, that is, by o¤ering a contract menu
to the informed party and letting him self-select, the present analysis shows that, unless
agent ability types are known to vary su¢ ciently, there is no point in agent discrimination.
This is so because agent ability types can be inferred ex post through the tournament and be
penalized or rewarded accordingly. By contrast, without tournaments, that is, with absolute
performance evaluation, high agent performance would be rewarded even if it were the sole
outcome of pure common luck!
A second, associated puzzle, is why principals do not discriminate agents even after
agent types are learned through repeated observation of performance (see, for instance,
Leegomonchai and Vukina (2005) who found no evidence of ex post agent discrimination
21
through a strategic allocation of varying quality inputs to the agents). But the analysis
can help provide a solution to that puzzle as well. The best application of our analysis is
the empirically observed lack of agent discrimination in broiler contracting through tournaments, even though tournaments have been used by processing companies (integrators) for
decades, yielding plenty of experience to them in order to …ne-tune the implementation of
tournaments. The solution to the seeming puzzle is that processors know that growers (i.e.,
chicken farmers) are su¢ ciently homogeneous to not warrant discrimination. Such discrimination would add to grower complaints about low pay and grower group composition risk
(see Tsoulouhas and Vukina 2001). It could also have detrimental e¤ects on agent e¤ort for
they would fear a rachet e¤ect if individualized contract o¤ers with more stringent terms
were made in the future following success. Besides, the strict requirements of relationshipspeci…c capital investments and adherence to strict production rules e¤ectively minimize the
expression of higher ability into signi…cantly and systematically higher performance.
22
Appendix A: Proof of condition (16)
Given that only ability types ai
e
a would accept the contract o¤er, the expected type of
R1
an agent who accepts the contract is ea ai f (ai )dai : Given that the population mean is ; it
follows that
Z
Z
1
However,
Z
e
a
ai f (ai )dai =
e
a
ai f (ai )dai :
1
e
a
ai f (ai )dai = lim
! 1
1
Z
e
a
ai f (ai )dai :
Integrating by parts, it follows that
Z
e
a
ai f (ai )dai =
Hence,
Z
e
a
1
ai f (ai )dai =
=
=
Z
e
a
0
ai F (ai )dai = e
aF (e
a)
e
aF (e
a) + lim
F( )
Z
Z
e
a
e
a
F (ai )dai :
F (ai )dai =
F ( ) + lim
! 1
! 1
Z ea
F (ai )dai =
e
aF (e
a) + lim 0 + lim
! 1
! 1
Z ea
F (ai )dai :
e
aF (e
a) + lim
! 1
23
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